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R E S E A R C H Open Access

Possible contribution of quantum-like

correlations to the placebo effect:

consequences on blind trials

Francis Beauvais

Correspondence:

beauvais@netcourrier.com

91, Grande Rue, Sèvres, France

Abstract

Background: Factors that participate in the biological changes associated with a

placebo are not completely understood. Natural evolution, mean regression,

concomitant procedures and other non specific effects are well-known factors that

contribute to the “placebo effect”. In this article, we suggest that quantum-like

correlations predicted by a probabilistic modeling could also play a role.

Results: An elementary experiment in biology or medicine comparing the biological

changes associated with two placebos is modeled. The originality of this modeling is

that experimenters, biological system and their interactions are described together

from the standpoint of a participant who is uninvolved in the measurement process.

Moreover, the small random probability fluctuations of a “real”experiment are also

taken into account. If both placebos are inert (with only different labels), common

sense suggests that the biological changes associated with the two placebos should

be comparable. However, the consequence of this modeling is the possibility for two

placebos to be associated with different outcomes due to the emergence of

quantum-like correlations.

Conclusion: The association of two placebos with different outcomes is

counterintuitive and this modeling could give a framework for some unexplained

observations where mere placebos are compared (in some alternative medicines for

example). This hypothesis can be tested in blind trials by comparing local vs. remote

assessment of correlations.

Keywords: Placebo effect, Quantum-like correlations, Experimenter effect,

Randomized clinical trials

Background

Much has been written about the “placebo effect”and the purpose of this article is not

to make a review on this topic [1–6]. In itself the term “placebo effect”is curious and

paradoxical. Indeed, as underscored by Moerman and Jonas: “The one thing of which

we can be absolutely certain is that placebos do not cause placebo effects. Placebos are

inert and don’t cause anything”[1]. For this reason, Ernst and Resch insisted to clarify

the definition of placebo by distinguishing “perceived placebo effect”and “true placebo

effect”[7]. Perceived placebo effect is the outcome that is associated with the placebo

group in a trial; it includes natural evolution of the disease, mean regression, concomi-

tant procedures and other non specific effects. True placebo effect is the difference

© The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International

License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium,

provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and

indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/

publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.

Beauvais Theoretical Biology and Medical Modelling (2017) 14:12

DOI 10.1186/s12976-017-0058-5

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

between perceived placebo effect and effect associated with no treatment. “No

treatment”groups are however infrequently performed and therefore there are often

some misunderstandings to define the scope of the “placebo effect”.

Since placebos are inert, the causes of the “true placebo effect”should be sought ra-

ther on the side of language and psychology. Thus, it has been shown that placebo ef-

fects can be caused by cognitive and emotional changes, expectation of symptom

changes or classical conditioning [2]. Actual effects of placebo on brain and body have

been evidenced and there are neurobiological underpinnings for these effects [8, 9].

However, in most studies aimed to decipher the placebo effect, patients are at the

centre of the investigations and all explanations rest on them. In the present article, the

experimental design and the experimenters are also taken into consideration. There-

fore, the focus is moved from patients to investigators and in this case the placebo

effect –at least one of its components –is not much different than an experimenter’s

effect. A famous example of experimenter’s effect was evidenced in the experiments of

Rosenthal et al. where an experimenter obtained from his subjects the data he expected

or wanted to obtain [10]. Outside of psychology, for example in cell biology or in physi-

ology, it is generally thought that such subtle influences could not be responsible for re-

sponse biases. In clinical trials, blind experiments are supposed to protect against any

outcome bias related to patient or physician; if such influences exist, they are distrib-

uted randomly in test and placebo groups. In the present article, it is suggested that

quantum-like correlations predicted by a probabilistic modeling could also contribute

to the “placebo effect”.

Results

Design of a minimal experiment with two placebos

The purpose of a typical experiment in medicine or biology is to establish a relationship

between a “cause”(independent variable) and a biological “effect”. Placebos (or

“controls”in experimental biology) are included in the experiment in order to assess

the effects of variables other than the independent variable.

We define a biological “object”(biological model or patients in a clinical trial) with

two possible states: no biological change (or resting state, not different from back-

ground noise) and biological change (“activated”state). A biological change may be de-

fined by setting a cut-off value of a continuous variable. We symbolize no biological

change as “↓”and biological change as “↑”.

We assume that all samples that are tested are placebos and that the only difference

is their labels which are either Pcb

0

or Pcb

1

. Since samples are all inert and physically

identical, common sense suggests that the biological outcomes associated with the two

placebos should be comparable. Nevertheless, the aim of the modeling is to know

whether in some circumstances the state “↑”could be more frequently observed with

one of the two labels (no matter which one at this stage). Therefore, the null hypothesis

(H

0

) of such an experiment is:

H0:Prob ↑jPcb0

ðÞ¼Prob ↑jPcb1

ðÞ ð1Þ

Prob (x∣y) is the conditional probability of xgiven y(or the probability of xunder

the condition y).

Beauvais Theoretical Biology and Medical Modelling (2017) 14:12 Page 2 of 17

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Figure 1 describes the two possible relationships between labels and biological out-

comes: either “direct”relationship (Pcb

0

associated with “↓”and Pcb

1

associated with

“↑”)or“reverse”relationship (Pcb

0

associated with “↑”and Pcb

1

associated with “↓”).

Note that this naming (“direct”or “reverse”) is arbitrary and does not prejudge results.

Of course, if a biological change is associated with the labels Pcb

0

and Pcb

1

with the

same rate (i.e. no relationship), then the probabilities of direct and reverse relation-

ship are both equal to 1/2. By convention, we present calculations of probability mainly

for direct relationship (the sum of the probabilities of direct and reverse relationships is

equal to one).

Description of an experiment from an uninvolved standpoint

The originality of the present modeling is that observers, observed system and their in-

teractions are described together from an uninvolved standpoint. The formalism is in-

spired from the relational interpretation of quantum physics [11, 12] and quantum

Bayesianism (QBism) [13, 14].

We suppose that the experimental landscape is described by a participant who is

uninvolved in the experiment. Suppose,asdescribedinFig.2,anobserverOwho

measures a variable of a system S; this variable can take one of the two values

“left”and “right”after a measurement. For a participant Puninvolved in the meas-

urement process, a definite value has been obtained after the measurement of Sby

O(either “left”or “right”). Pknows that O has observed a defined value after

measurement, but Pdoes not know what O has observed. If Pfinally observes the

system S, he records a definite value and he agrees with Oon this value when P

and Ointeract. Interactions between observers are like measurements and they

allow establishing correlations.

In this last case, it is important to underscore that it is not correct to say that Pis

“forced”to observe what Oobserved before they interact. Indeed, one can imagine an-

other participant Qwho in turn describes S,Oand Pwithout interacting with them.

What Qcan say is that a correlation has been established between S,Oand P, but Q

Fig. 1 Relationships between placebos and biological system. There are two possible placebos (“0”and “1”)

and two possible states for the biological system: no change (“resting”state or background) which is noted

“↓”and biological change above background (“activated”state) which is noted “↑”. As a consequence, there

are two possible relationships defined as: direct relationship with “placebo 0”associated with “↓”and

“placebo”1 associated with “↑”; reverse relationship with “placebo 0”associated with “↑”and “placebo 1”

associated with “↓”

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cannot say which result is observed. The only thing that an uninvolved participant can

do is to describe the form (correlations), but not the content (outcomes) of the informa-

tion available to the observers who interact with Sand with each other. Thus, the

consistency of any measurement is guaranteed.

Note that, strictly speaking, an uninvolved participant does not describe the “reality”

itself (made of “contents”), but he constructs a predictive tool (made of correlations

and probabilities) in order to know what to expect if he decides to interact with the

“real”observers.

Probabilistic modeling of a minimal experiment with two placebos

An experiment is modeled from the standpoint of a participant Pwho is outside the la-

boratory, as described in the previous section. The participant Pdoes not interact with

the “objects”that he describes and he remains uninvolved in the evolution of the ex-

perimental situation. The role of Pis to describe the evolution of a team of interacting

experimenters with the knowledge of the initial conditions.

We consider a team composed of two experimenters named Oand O’who observe

the biological system S. We suppose an experimental situation where the probability

for each experimenter to observe a direct relationship (as defined in Fig. 1) is pand the

probability of a reverse relationship is q(with p+q= 1).

Each observer has his own probabilistic expectations and the uninvolved participant

Passigns the probability pto Oas the best estimate that Ocan make for the future ob-

servation of the direct relationship. The same probability pis assigned to O’independ-

ently of Osince the probabilistic expectations are specific to each observer.

Fig. 2 Description of an experiment from an uninvolved standpoint. The observer Omeasures the system

Swhereas the participant Premains uninvolved in the measurement (he does not interact with Oand S).

Pknows that Ohas observed a definite state of S, but he does not know which one. If Pfinally interacts

with Oand S, then Pand Oagree on the outcome of S. The reasoning can be continued with another

participant Qwho does not interact with S,Oand P. What Qcan say is that S,Oand Phave definite values

that are correlated. The only thing that an uninvolved participant can do is to describe the form, but not

the content of the information available to the observers who interact with Sand with each other. Thus,

the consistency of any measurement is guaranteed (GNU Free Documentation License)

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Therefore, the participant Pconstructs a predictive tool for his own use where the ex-

perimenters have independent probabilistic expectations on the experimental outcome

and are in agreement when they compare their records. The future outcome that Oex-

pects to record (event A) and the future outcome that O’expects to record (event B)

are independent events in the probability space constructed by P(Fig. 3). This condi-

tion of independence is easily formalized since the probabilities of two independent

events Aand Bhave well-known mathematical properties:

Prob A∩BðÞ¼Prob AðÞProb BðÞ ð2Þ

Therefore, when Oand O’interact and agree on the result of the experiment (i.e. the

events of the set A∩B), the best estimate of the probability that Oand O’observe a

direct relationship is Prob (direct)=p×paccording to Eq. (2) since the probability to

record a direct relationship was estimated to be pfor Oand also pfor O’(Fig. 3). Simi-

larly, the best estimate of the probability that Oand O’observe a reverse relationship is

Prob (reverse)=q×q.

The intersubjective agreement discards some impossible situations such as Oobserves

a direct relationship while O’observes a reverse relationship. Since the sum of the prob-

abilities of all possible events is equal to one, Prob (direct)=p×pmust be renormalized.

For this purpose, p×pis divided by the sum of the probabilities of all possible outcomes

(grey areas in Fig. 3), namely direct relationship (p×p) and reverse relationship (q×q):

Prob directðÞ¼

p2

p2þq2ð3Þ

Fig. 3 Probabilistic space constructed by an uninvolved participant Pto predict the outcomes of the

experiments. A team of interacting experimenters Oand O’is described from the standpoint of an uninvolved

participant who knows the initial experimental conditions (Fig. 2). We suppose a probability equal to pfor the

event “direct relationship”and equal to qfor the event “reverse relationship”(p+q= 1). Each observer has his

own probabilistic expectations and Passigns the probability pto Oas the best estimate that Ocan make for

the future observation of a direct relationship; the same probability is assigned to O’independently of Osince

the probabilistic expectations are specific to each observer. White areas are unauthorized experimental

situations with incompatible outcomes after interaction of Oand O’(e.g. “direct”for Oand “reverse”for O’).

Therefore, the probability that the experimenters observe a direct relationship is calculated by dividing the

central gray area (“direct”for both observers) by the sum of the probabilities of possible outcomes (either

“direct”or “reverse”for both observers), namely all gray areas. Ω, probability space

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By dividing both the numerator and the denominator by p

2

(taking into account that

p+q= 1), the only variable of the equation is p:

Prob directðÞ¼ 1

1þ1

p−1

2ð4Þ

Eqs. (3), (4) and Fig. 3 are easily generalized to any number Nof observers who all

agree on the outcome:

Prob directðÞ¼ 1

1þ1

p−1

Nð5Þ

In a “real”experiment, particularly in biology, random fluctuations occur and they

must be taken into account because, after each elementary fluctuation, a tiny bias is in-

troduced and Prob (direct) must be updated.

In the next lines we calculate the evolution of the probability for Oand O’to observe a

direct relationship from the standpoint of the participant P. First, we write that Prob

(direct) is equal to 1/2 in the absence of observers (N= 0 in Eq. (5)). As a consequence,

the initial value of Prob (direct)attimet

0

before the first fluctuation is equal to p

0

=1/2.

We then introduce ε

i

as successive elementary random fluctuations of Prob (direct)

that occur during successive elementary intervals of time (ε

i

are positive or negative

real random numbers such as ∣ε

i

∣< < 1). Note that an implicit consequence of the ran-

dom fluctuations of Prob (direct) is a non-null, but very small, probability to observe a

biological change (“↑”).

After the first fluctuation ε

1

, we easily calculate with Eq. (4) the updated probability

p

1

which is based on p

0

+ε

1

. The equation is then generalized for any probability p

n+1

based on previous probability p

n

and fluctuation ε

n+1

. We obtain a mathematical se-

quence which allows calculating the successive probabilities of a direct relationship:

Probnþ1ðdirectÞ¼pnþ1¼1

1þ1

pnþεnþ1−1

Nwith p0¼1=2ð6Þ

Two placebos associated with different outcomes

Equation 6 allows calculating the successive states of a system constituted of a bio-

logical system and a team of interacting experimenters/observers committed in the es-

tablishment of a supposed relationship.

A computer calculation of this mathematical sequence is described in Fig. 4 after 100

successive random fluctuations ε

i

(with values around 10

−15

) and with two observers

(N= 2). We observe that the initial situation is in fact metastable if fluctuations are

taken into account. Indeed, in all cases (i.e. whatever the series of values ε

i

), a dramatic

transition towards one of two stable positions is achieved:

Prob ðdirectÞ¼1=2ðmetastable positionÞ

↓

Prob ðdirectÞ¼1or 0ðtwo possible stable positionsÞ

ð7Þ

All samples of an experiment are thus engaged either in a direct relationship or in a

reverse relationship. Note that the probability of a biological change was very small

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initially and, after the transition, a biological change is systematically associated with

the label Pcb

1

in stable position #1 or systematically associated with the label Pcb

0

in

stable position #2. The choice of one of the two stable positions is random. In both

cases, a relationship (direct or reverse) between labels and outcomes emerges.

However, the purpose of an experiment is to compare a “test”situation with a

“control”situation. Biological systems are therefore prepared in an asymmetrical state

with resting state (background noise) implicitly associated with a “control”situation.

The stability of the resting state (or basic state) is a condition for a proper assessment

of the samples (the experiment begins with the preparation of the biological system be-

fore samples are tested). In other terms, the state of the biological model at rest (before

each test) can be considered as associated with the label “control”.

We suppose, for example, that Pcb

0

is considered as a “control”by the experimenters.

Consequently the stable state #2 eliminates itself since Pcb

0

cannot be associated both

with change (when Pcb

0

samples are tested) and with no change (for the resting state).

Only the stable position #1 is a possible state:

Prob ðdirectÞ¼1=2ðmetastable positionÞ

↓

Prob ðdirect Þ¼1ðstable positionÞ

ð8Þ

A probability equal to one for the direct relationship means that the participant Pis

assured –if he finally interacts with the team of experimenters after the end of the

experiment –to observe a direct relationship between labels and biological outcomes.

Thanks to probability fluctuations, a biological change associated with each sample

with Pcb

1

label emerges from background noise.

Fig. 4 Calculation of the probability of a direct relationship. The evolution of the probability that a team

(composed of two members who interact) observes a direct relationship is described in this figure by taking

into account successive probability fluctuations. The probability defined in Fig. 3 is calculated according to

the mathematical sequence in the cartouche. Each successive probability p

n+1

of the sequence is calculated

by using p

n

and a random probability fluctuation is randomly obtained between −0.5 and +0.5 × 10

-15

. This

computer simulation shows that the initial state with a probability of 1/2 is in fact metastable and, after a

dramatic transition, one of two stable positions is achieved: either Prob (direct) = 1 or Prob (direct) = 0. With

N> 2 or with higher values of probability fluctuations, a transition is obtained after a lower number of

calculation steps (data not shown). Eight computer simulations are reported in this figure

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Consequences of different types of blind designs on correlations

Until now we examined an experimental situation where the observers Oand O’

assessed themselves the rates of correlation between labels and biological outcomes

(open-label experiments). Nevertheless, the labels can be masked to the experi-

menters in order to reduce or eliminate any bias. After the outcomes have been

obtained in blind conditions for all samples, the labels of samples are unmasked.

In this article, we distinguish blind experiments with either local or remote assess-

ment of correlations.

For a local assessment of correlations with blind design, an automatic device or a

member of the team of experimenters keeps secret the labels of the samples until the

end of the experiment. In this case, the automatic device or the observer who is dedi-

cated to the blinding are also elements of the experiment because they interact with

the other observers and can be described (from the standpoint of P) with the same

modeling as open-label experiments.

A remote assessment of correlations with blind design is typically used in randomized

clinical trials (also named centralized blind design). The remote supervisor (a statisti-

cian for example) does not interact with the experimenters before all measurements are

completed. It is important to underscore that the remote supervisor should not be con-

fused with the uninvolved participant Pwho describes the experiment. Indeed, Pdoes

not interact and is not involved in the experiment. With a remote supervisor, the ex-

perimenters observe biological outcomes, but have no feedback on labels before the re-

mote experimenter is aware of the rate of success. As a consequence, Prob (direct)=

Prob (reverse); since Prob (direct) + Prob (reverse) = 1, then Prob (direct) = 1/2. In

summary:

Prob (direct) = 1 with local assessment of correlations;

Prob (direct) = 1/2 with remote assessment of correlations.

Figure 5 illustrates the consequences of the assessment of the correlations with a re-

mote assessment according to the modeling. In this case (blind experiment with an ex-

ternal supervisor), there is no statistical difference between the biological outcomes

associated with Pcb

0

and Pcb

1

in contrast with a local assessment (local blind design or

open-label experiment).

The experimental context is therefore crucial for establishing a relationship in

the modeling. With a local assessment, the experimenters observe labels and then

biological outcomes (open-label experiment) or observe biological outcomes and

then labels (local blind experiments). In contrast, with a remote supervisor, the ex-

perimenters observe biological outcomes, but have no feedback on labels. If a local

observer/experimenter is the first to assess the relationship, correlations emerge; if

a remote supervisor is the first to assess the relationship, correlations vanish (bio-

logical changes are nevertheless observed, but at random places). Of course, in all

cases, when participants met together, they agree on the conclusion (correlation or

no correlation). The order of the assessments (local first or remote first) is the key

element for the degree of correlation.

It is important to underscore that this difference between local and remote assess-

ment of correlations offers the opportunity to test the modeling.

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Characterization of the role of the observers O and O’

The experimenters/observers Oand O’play a crucial role in the modeling and we

examine in this section how their involvement could be characterized and quantified.

As previously reported in Eq. (2), the joint probability of two independent events A

and Bis equal to the product of the separate probabilities of Aand B. This equation

can be generalized for two events Aand Baccording to their degree of independence:

Prob A∩BðÞ¼Prob AðÞProb BðÞþdwith 0 ≤d≤1ðÞ ð9Þ

The degree of independence increases when the value of ddecreases; the two events

are completely independent with d= 0. In other words, the correlation of the two

events increases when the value of dincreases. Eq. (3) can be easily modified if dis

taken into account (Fig. 6; see legend for calculation details):

Prob directðÞ¼

p2þd

p2þq2þ2dwith 0 ≤d≤pqðÞ ð10Þ

When the parameter dvaries from d=pq to d= 0, the experimental situation pro-

gressively shifts from a classical description to the present modeling (Fig. 6).

“Observing”an experiment requires a frame (what are we expecting?) and a feedback

(what did we record?). Equation (6) indicates that there is no transition of Prob (direct)

towards a stable position in the absence of observers (N= 0). We can draw the same

conclusion if the observers are physically present in the laboratory, but not focusing

Fig. 5 Comparison of local vs. remote assessment in an experiment with two placebos. In an experiment with

a local assessment (local blind design or open-label experiment), correlations between labels (Pcb

0

and Pcb

1

)

and states of the biological system (↓and ↑)emerge(band e) from the initial state (aand d). These correlations

vanish if the assessment of the experiment is made in a blind experiment with a remote supervisor (cand f). In

this latter case, the difference between the biological changes associated with Pcb

0

and Pcb

1

is not statistically

significant (NS) and the biological changes (“↑”) are randomly distributed among the two placebos. The

difference of results in local vs. remote assessments offers the opportunity to test the modeling

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their attention on this specific experiment (they expect nothing about the experimental

system and they do not receive feedback). As a consequence, the parameter dcan be

considered as an evaluation of the attention of the team of experimenters to observe

the predefined relationship between labels and biological outcomes. When d= 0, the

observers are fully committed and for d=pq their attention is completely drawn away

from the experiment. For intermediate values, the team is more or less occupied with

these observations. Therefore, experimenters’qualities, such as attention, commitment

and persistence, appear to be necessary during the experiments for the emergence of

correlations between labels and biological outcomes.

We can go a step further by considering that the parameter dis also an assess-

ment of the capability of the experimenters to recognize (or not) the direct and re-

verse relationships per se, i.e. as new “objects”regardless their components, namely

the association of the biological outcomes with Pcb

0

and Pcb

1

. Indeed, in Eq. (4)

(i.e. with d= 0), it is implicit that the experimenters recognize the outcome per se

(i.e. in its “wholeness”or as such) as it would be the case for the outcome of a

dice roll or the position of a pointer on a measurement device. But suppose now a

team of experimenters Oand O’who are inexperienced and do not recognize the

predefined experimental relationship as a structured ensemble. The experimenters

identify the sub-events as separate elements without integrating them as a whole

(these sub-events are the association of the biological outcomes with Pcb

0

and

Pcb

1

). Since we continue to adopt the standpoint of P, we use Eq. (4) to calculate

the evolution of the probability of each sub-event. Before the first fluctuation prob-

ability, the probabilities of the two sub-events are: Prob (direct |Pcb

0

)=1 and Prob

(direct |Pcb

1

)=0(seeFig.5a).WenoticethatProb(direct |Pcb

0

) and Prob (direct |Pcb

1

)

are already in stable positions. Therefore, by using Eq. (4) (see also Fig. 4), these condi-

tional probabilities are maintained in their respective stable positions with Prob (direct |

Pcb

0

) that tends toward 1 and Prob (direct |Pcb

1

) that tends toward 0. The experimental

Fig. 6 From a classical description of the experimental situation to the present modeling. The experimental

situation depicted in Fig. 3 is generalized in this figure by using the parameter dwhich varies with the

degree of independence of the probabilistic expectations on the outcome assigned to Oand O’. The values

of the two areas with impossible situations (direct relationship for one observer and reverse relationship for

the other one) are calculated as: p–(p

2

+d)=p×(1 –p)–d=pq –d. For d= 0, correlations between labels

and biological outcomes emerge and, for d=pq, the probability of a direct relationship is equal to pas in

classical probability. Ω, probability space

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results associated with these uncoupled sub-events can be gathered in order to calculate

Prob (direct) by using the law of total probability:

Prob directðÞ¼Prob Pcb0

ðÞProb directjPcb0

ðÞþProb Pcb1

ðÞ

Prob directjPcb1

ðÞ ð11Þ

¼1=21þ1=20¼1=2ð12Þ

As a consequence, because the relationship between labels and biological outcomes is

not recognized as a structured ensemble, there is no transition and Prob (direct) tends

toward 1/2.

These considerations are reminiscent from Gestalt psychology that states that human

mind spontaneously tends to perceive phenomena as structured ensembles (Gestalt)

and not as a simple addition of parts. For example, the well-known Necker cube is im-

mediately recognized as a 3D cube by a human observer and not as the simple addition

of lines drawn on a 2D sheet [15]. We instantly “see”a cube in space because we have

learned to perceive these 2D drawings as 3D “objects”.

As for Necker cube, cognitive and learning processes are undoubtedly at work for the

passage from an “analytic”(d=pq)toa“structured”(or global) perspective (d= 0). In

the first situation (d=pq), the experimenters are spectators of the experimental landscape

that is perceived as a “collection of points”;inthesecondsituation(d= 0), they are actors

who interpret the experimental landscape that is perceived as a “form”[15]. In this last case,

the experimenters concentrate their attention towards a “transcendent object”(namely, the

predefined relationship) without reference to the details that become indiscernible.

If cognitive and learning processes are involved in the emergence of quantum-like

correlations, different teams of experimenters with different training and experience

should report various degrees of correlations between labels and biological outcomes in

experiments comparing two placebos.

Emergence of a quantum-like logic

Only tools from classic probability are used in the modeling. Nevertheless, as demon-

strated in this section, there is an underlying quantum-like logic which is rooted in the

initial partition of placebos as Pcb

0

and Pcb

1

. Indeed, according to Fig. 1:

Prob Pcb0

ðÞProb ↓ðÞþProb Pcb0

ðÞProb ↑ðÞþProb Pcb1

ðÞProb ↓ðÞ

þProb Pcb1

ðÞProb ↑ðÞ¼1:

ð13Þ

When the stable position #1 is achieved, Prob (Pcb

0

) = Prob (↓) and Prob (Pcb

1

)=

Prob (↑) (see Fig. 5b); when the stable position #2 is achieved, Prob (Pcb

0

) = Prob (↑)

and Prob (Pcb

1

) = Prob (↓). Therefore in both cases:

Prob Pcb0

ðÞ½

2þProb Pcb1

ðÞ½

2þ2Prob Pcb0

ðÞProb Pcb1

ðÞ¼1ð14Þ

This equation is equivalent to:

Prob Pcb0

ðÞþProb Pcb1

ðÞ½

2¼1ð15Þ

Then, we define aand bsuch as Prob (Pcb

0

)=a

2

(or a.a)andProb(Pcb

1

)=b

2

(or b.b). These definitions correspond to the stable position #1 (for the stable pos-

ition #2, b

2

must be taken equal to –b×−b):

Beauvais Theoretical Biology and Medical Modelling (2017) 14:12 Page 11 of 17

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

a⋅aþb⋅bðÞ

2¼a⋅aðÞ

2þb⋅bðÞ

2þ2a⋅bðÞ

2¼1ð16Þ

a⋅aþb⋅bðÞ

2þb⋅a−a⋅bðÞ

2¼a⋅aðÞ

2þb⋅bðÞ

2þb⋅aðÞ

2þa⋅bðÞ

2¼1ð17Þ

1þ0¼1=2þ1=2¼1ð18Þ

As can be seen in Fig. 7, the left-hand side of Eq. (17) is the sum of Prob (direct) plus

Prob (reverse) without a remote supervisor and the right-hand side is the sum of Prob

(direct) plus Prob (reverse) with a remote supervisor. The terms aand bare thus

probability amplitudes and their squaring allows calculating the corresponding

probabilities.

Therefore, the probability of a direct relationship without a remote supervisor is cal-

culated by doing the sum of the probability amplitudes of the two paths that lead to a

direct relationship and then by squaring this sum. With a remote supervisor, the prob-

ability of a direct relationship is calculated by squaring the probability amplitude of

each path that leads to a direct relationship and then by making the sum of the prob-

abilities of the two paths (Fig. 7).

The relationship between labels and biological outcomes in the modeling has the

same logic as single-photon self-interferences in Young’s double-slit experiment where

photons behave either as particles when paths are detected or as waves when paths are

not detected. In Fig. 7 that sketches an elementary experiment, quantum-like correla-

tions are observed when “paths”(i.e. labels) are undistinguishable (from an outside

standpoint) and correlations vanish when they are distinguishable for a remote super-

visor. In this last case, each label is forced to adopt a defined “pathway”.

The emergence of quantum-like correlations is the consequence of the initial as-

sumptions, namely the independent probabilistic expectations and the intersubjective

agreement. The concomitant consideration of these two assumptions implies that the

outcome of an experiment does not pre-exist to the interaction of Oand O’from the

standpoint of P. This is a characteristic of quantum measurements and, in the language

of quantum mechanics, the “state”of Oconcerning his identification of the outcome

Fig. 7 Probability of a direct relationship without or with a remote supervisor. The quantum-like probability

of a direct relationship is calculated as the square of the sum of the probability amplitudes of the different

possible “paths”. With a remote supervisor, classical probability applies and the probability of a direct

relationship is calculated as the sum of squares of the probability amplitudes of the “paths”. Therefore, the

probabilities of a direct relationship are different without or with a remote supervisor

Beauvais Theoretical Biology and Medical Modelling (2017) 14:12 Page 12 of 17

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

would be said “superposed”before interacting with O’(and vice versa). The intersub-

jective agreement plays a similar role as a conservative law in physics and Oand O’

would be said “entangled”after their interaction.

Importance of an uninvolved standpoint

The uninvolved standpoint of the participant Pis central in the construction of the

modeling. Indeed, from the standpoint of O, if he observes a direct relationship or a re-

verse relationship, then he can hold for sure that O’will tell him that he observes the

same event. As a consequence the probability that Oand O’observe a direct relation-

ship is pin this case as stated by classical probability and not p×p(before

renormalization) from the standpoint of P. The standpoints of Pand O-O’coincide in

situations where these two equations are verified:

p¼p2

p2þq2and q¼q2

p2þq2ð19Þ

These two equations are equivalent to (2p–1) (p–1) = 0 and (2q–1) (q–1) = 0, re-

spectively. Therefore, there are only three possible values for p: 1/2, 1 or 0. These

values are the probabilities associated with initial position, stable positions #1 and #2,

respectively. Only the outside standpoint of Pwho is not involved in the observation of

the experiment allows describing the transition of Prob (direct) from 1/2 to 1 (or 0) as

a consequence of the emergence of quantum-like “interferences”(i.e. the cross-terms

with probability amplitudes equal to band -b in Fig. 7).

The differences between the standpoints of O-O’and Pare the consequence of the

demonstration of Breuer about the impossibility of a complete self-measurement [16].

According to this demonstration, a measurement apparatus (or an observer) is unable

to distinguish all the states of a system in which it is contained (whether this system is

classical or quantum mechanical does not matter). Only a second external apparatus

(P) that observes both the first apparatus (O) and the system (S) is able to account all

correlations between Oand S[17].

Optimized placebos in clinical trials

Without any doubt, the success of many complementary or alternative medicines rests

on placebo effect. Thus, most authors consider homeopathy as a perfect illustration of

the enforcement of the placebo effect in medicine. Moreover, homeopathic medicines

could be considered as “super placebos”(or optimized placebos) since even practi-

tioners think that they prescribe “true”medicines despite the absence of active mole-

cules. Indeed, the manufacturing process of a majority of homeopathic medicines

eliminates the initial active molecules by serially diluting them well beyond the limit set

by Avogadro’s number. In other words, there are zero active molecules in these highly

diluted samples. Even if tiny traces of the initial molecules would be present (due to

contamination or imperfect diluting process), it remains to demonstrate how they

could nevertheless have an effect contradicting the law of mass action.

Since no classical pharmacological action can be assigned to high dilutions, it has

been suggested that modifications of water structure during the dilution process could

account for the alleged effects. Until now, no convincing evidence has been reported

indicating that modifications of water structure specific of the initial molecules are able

Beauvais Theoretical Biology and Medical Modelling (2017) 14:12 Page 13 of 17

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

to induce specific biologic changes. Moreover, homeopathy medicines available in phar-

macies are sugar granules that have been impregnated with high dilutions and then

dried. Therefore, until there is evidence to the contrary, the most reasonable scientific

attitude is to consider homeopathy medicines and high dilutions as plain placebos.

Interestingly, the gold standard for drug evaluation, namely blind randomized clinical

trial, appears to be an obstacle in studies aimed to establish the efficacy of homeopathy

medicines. Thus, the study of Shang et al. compared homeopathy trials and matched

conventional-medicine trials [18, 19]. The authors concluded that homeopathy medi-

cines were comparable to placebos. Indeed, in contrast with conventional medicines,

double-blind design was associated with a strong decrease of the probability of success

when compared with open-label design. Although this study has been heavily criticized

by proponents of homeopathy, most of them nevertheless acknowledge that blind ran-

domized clinical trials are not adequate for assessing homeopathy medicines [20, 21]. A

randomized clinical trial by Brien et al. in patients with rheumatoid arthritis suggested

that homeopathy consultations, but not homeopathy medicines, were associated with a

clinical benefit thus reinforcing the idea of a placebo effect [22].

In 2013, I proposed a slight modification of trial design in order to increase the

chance to observe a difference between outcomes in double-blind placebo-controlled

randomized trials of homeopathy medicines. This suggestion was not an encourage-

ment for the practice of homeopathy, but an attempt to understand the persisting suc-

cess of this alternative medicine in the absence of a rational basis. Based on the

hypothesis that quantum-like correlations were responsible for “successful”open-label

homeopathy clinical trials, it was proposed to replace the centralized assessment of effi-

cacy in blind trials (generally done by statisticians) with a local assessment (by physi-

cians) [23]. Thieves et al. recently challenged this hypothesis and reported experiments

in a plant model (wheat germination) that compared a homeopathy medicine and a pla-

cebo both in local and centralized blind designs [24]. The results were in favor of the

initial hypothesis since a significant difference of plant growth was observed between

homeopathy medicine and placebo with local assessment while there was no significant

difference with centralized assessment. The interaction test for local vs. centralized

blind designs was statistically significant (p = 0.003). If we consider all samples (includ-

ing homeopathy medicine) as plain placebos that differ only by their labels, these re-

sults are in favor of the present hypothetical modeling. These results should be also an

encouragement for physicians to implement the same local blind design in clinical trials

comparing a placebo with homeopathy medicine (i.e. a second placebo) in order to test

in vivo the hypothesis of quantum-like correlations as depicted in Fig. 5.

Discussion

It is generally thought that the macroscopic world escapes to the consequences of

quantum physics due to the decoherence process. As a consequence, biological systems

are considered to behave only classically. Nevertheless, some phenomena such as

photosynthetic light harvesting or avian magnetoreception have been recently sug-

gested to be the consequence of quantum phenomena [25]. Asano et al. evidenced

quantum-like probabilistic behavior in Escherichia coli lactose-glucose metabolism [26].

In experimental psychology, some processes of cognition appear to obey to nonclassical

logic [27]. Thus, the purpose of the new field named "quantum cognition”is to describe

Beauvais Theoretical Biology and Medical Modelling (2017) 14:12 Page 14 of 17

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

cognitive processes such as reasoning, decision making, judgment, language, memory

or perception with mathematical quantum tools [28–31]. Moreover, Aerts described

some experimental situations in physics where macroscopic devices could exhibit a

quantum-like behavior [32]. Interestingly, Aerts showed that quantum probabilities

could be introduced as the consequence of a lack of knowledge about fluctuations dur-

ing the interaction between a measuring device and the object to be measured [32].

Most authors that use quantum probability outside the field of physics do not consider

that the systems they describe are really quantum. Tools of quantum probability are

simply used to describe results that until then were considered paradoxical [27]. In-

deed, quantum physics is not only a new mechanics but also a new probability theory.

An extension of classical probability with some mathematical tools borrowed to

quantum probability (e.g. superposition, entanglement, interferences) appears to be

fruitful in these different domains. With the present hypothetical modeling, it is pro-

posed that quantum-like correlations could be a component of the placebo effect.

A central question is the generalisability of the proposed modeling to other experi-

mental situations. Indeed, one could argue that bets on a coin toss could be also de-

scribed by the same modeling by replacing labels with bets and biological system with

coin toss. The answer is in Eq. (6) that supposes first that the system Shas an internal

structure submitted to small random fluctuations (thermal fluctuations for example)

and second that each p

n+1

value is strongly dependent on p

n

value. In other words,

probabilities p

n+1

are correlated with probabilities p

n

. This last characteristic is named

temporal autocorrelation and is a feature of phenomena with slow random fluctuations

such as systems submitted to Brownian motion or biological systems. Of course, an-

other implicit condition is the absence of physical obstacles that would block the transi-

tion of Prob (direct). Therefore, for systems based on a phenomenon not submitted to

internal fluctuations (radioactive decay) or “rigid”systems with sufficient mechanical

inertia to be not influenced (coin flipping or dice rolling), εis equal to zero and no

transition is possible. For experimental systems submitted to internal fluctuations, but

with successive states that are not autocorrelated due to strong restoring forces

(“elastic”systems), a transition as described in Fig. 4 is not possible (only random fluc-

tuations around 1/2 are observed). An example of such a system is a beam splitter that

randomly transmits or reflects a photon and vibrates around a fixed point. Systems

based on phenomena with large random fluctuations (electronic noise for example) are

also unsuitable.

The possibility to be experimentally tested is the hallmark of a scientific theory.

The proposed modeling predicts that quantum-like correlations vanish when they

are assessed by a remote supervisor. Only local assessments allow quantum-like

“interferences”with correlation of “expected”and observed outcomes. It is import-

ant to emphasize that this modeling does not describe a causal relationship be-

tween mental states (e.g. intention) and physical states. Indeed, only quantum-like

correlations are allowed and there is no way to transmit messages, instructions or

orders from a laboratory to another one by using a series of coded samples.

Walach has extensively studied the relationship between homeopathy and notions

from quantum logic such as complementarity and entanglement by using a “generalized

quantum theory”[33, 34]. Of interest, this author insisted that homeopathy medicines

and their associated clinical outcomes could not be treated causally (as it the case in

Beauvais Theoretical Biology and Medical Modelling (2017) 14:12 Page 15 of 17

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

blind randomized clinical trials), otherwise mismatches between outcomes occurred

[35]. The present modeling with two placebo, which are differently labeled, leads to the

same conclusion. Moreover, it is not excluded that quantum-like correlations could

emerge in clinical trials for conventional drugs and add to classical causal relationship.

Some authors reported clinical trials where placebos associated with different labels

or therapeutical rituals could lead to different outcomes [36, 37]. Only psychological

mechanisms were supposed to be the cause of the different outcomes. Nevertheless, it

would be interesting to evaluate a possible involvement of quantum-like correlations in

such experiments aimed at investigating the placebo effect.

The potential existence of quantum-like correlations in the context of the experi-

menter effect could be also an element interesting to explore in the current debate

about low reproducibility in life sciences [38]. Indeed, differences among experimenters’

teams are expected for the establishment of quantum correlations according to the

modeling. As a matter of fact, trials in biology, medicine or psychology could benefit

from an extended theory of probability that permits interferences between probabilities

(more exactly between probability amplitudes).

Conclusion

The hypothetical modeling proposed in this article suggests that two placebos with dif-

ferent labels can be associated with different outcomes even in blind trials. Such a

counterintuitive conclusion is the consequence of a probabilistic modeling that autho-

rizes quantum-like interferences. This modeling could give a framework for some unex-

plained observations where mere placebos are compared (in some alternative

medicines for example) and could be tested in blind trials by comparing local vs. re-

mote assessment of correlations.

Acknowledgements

Not applicable.

Funding

No funding

Availability of data and material

Not applicable.

Authors’contributions

Not applicable (unique author)

Competing interests

The author declares that he has no competing interests

Consent for publication

Not applicable.

Ethics approval and consent to participate

Not applicable.

Publisher’sNote

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 15 February 2017 Accepted: 25 May 2017

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2.

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4.

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